{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R934",
  "content_hash": null,
  "slug": "binary-word-logarithmic-dfa-separation-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. A March 2025 primary preprint proves \\(S(n)=O(\\log^2 n)\\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound. Prove that \\(s(x,y)\\le C\\log n\\) for an absolute \\(C\\), every sufficiently large \\(n\\), and all distinct binary \\(n\\)-letter words, or give infinitely many pairs with \\(s(x,y)/\\log n\\) unbounded.",
  "relevance": "For binary word logarithmic dfa separation, pins the dated research frontier: UNKNOWN as of 2026-07-31. A March 2025 primary preprint proves \\(S(n)=O(\\log^2 n)\\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound. Prove that \\(s(x,y)\\le C\\log n\\) for an absolute \\(C\\), every sufficiently large \\(n\\).",
  "relevance_source": "recorded",
  "body": "UNKNOWN as of 2026-07-31. A March 2025 primary preprint proves \\(S(n)=O(\\log^2 n)\\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.\n\nA complete resolution must satisfy this condition: Prove that \\(s(x,y)\\le C\\log n\\) for an absolute \\(C\\), every sufficiently large \\(n\\), and all distinct binary \\(n\\)-letter words, or give infinitely many pairs with \\(s(x,y)/\\log n\\) unbounded.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2503.23184",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2503.23184",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "relations": [
    {
      "slug": "R933",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "binary-word-logarithmic-dfa-separation",
      "title": "binary word logarithmic dfa separation",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}